Phase coexistence at the first-order Mott-transition revealed by pressure-dependent dielectric spectroscopy of $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$

The dimer Mott insulator $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ can be tuned into a metallic and superconducting state upon applying pressure of 1.5 kbar and more. We have performed dielectric spectroscopy measurements (7 kHz to 5 MHz) on $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ single crystals as a function of temperature (down to $T=8$ K) and pressure (up to $p=4$ kbar). At ambient conditions, a relaxor-like dielectric behavior develops below 50 K that shifts toward lower temperatures as the crystal is pressurized. Interestingly, a second peak emerges in $\varepsilon_{1}(T)$ around $T=15$ K, which becomes strongly enhanced with pressure and is attributed to a small volume fraction of metallic puddles in the insulating host phase. When approaching the phase boundary, this peak diverges rapidly reaching $\varepsilon_{1} \approx 10^{5}$. Our dynamical mean-field theory calculations substantiate that the dielectric catastrophe at the Mott transition is not caused by closing the energy gap, but due to the spatial coexistence of correlated metallic and insulating regions. We discuss the percolative nature of the first-order Mott insulator-to-metal transition in all details.